Small Forcing Creates Neither Strong nor Woodin Cardinals

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals. The widely known Levy-Solovay Theorem [LevSol67] asserts that small forcing does not affect the measurability of any cardinal. If a forcing notion P has size less than κ, then κ is measurable in V P if and only if it is measurable in V ; every measure on κ in V extends uniquely to a measure on κ in V ; the corresponding embeddings lift uniquely from V to V , and all the measures in V P arise in this way. The argument generalizes to show the same fact for small forcing with the other large cardinals which are witnessed by the existence of certain kinds of ultrapowers, such as strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on. Missing, however, from the scope of the theorem are both the strong cardinals and the Woodin cardinals, whose embeddings are not simple ultrapowers, but directed systems of them. Several years ago the second author of this paper successfully treated these extender embeddings and proved that small forcing creates neither strong nor Woodin cardinals. Since his proof seems not to be widely known, we would like to present it here along with a level-by-level version which aims to obtain the result for cardinals which are only partially strong. Main Theorem. After small forcing, a cardinal κ is strong if and only if it was strong in the ground model.